 Solution

ID: past-exam-of-the-algebraic-curves-course-of-the-university-of-oxford/2023/3/c/solution

The generator

η = f d h

(η)

is an effective divisor, since there are no poles. In genus one, the degree of a divisor formula in part (a) gives

de g (η) = 2 g - 2 = 0.

An effective divisor of degree zero has every coefficient zero. Thus

(η) = 0, f d h has no zeros anywhere on C .

This proves nonvanishing of holomorphic differentials in genus one. It concerns the differential itself: a zero or pole of the particular coefficient

f

may be cancelled by the local behavior of

d h

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